X184

INVERSE OF X125 IN THE BROCARD CIRCLE

X184 is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B1 and C1 be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C2, A2; A3, B3. Then A'B'C' is formed by the perpendicular bisectors of segments B1C1, C2A2, A3B3. (Fred Lang, Hyacinthos #1190)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 6, 25 }: Central Line L(6, 25) { 3, 49 }: Central Line L(3, 49) { 4, 54 }: Central Line L(4, 54) { 26, 52 }: Central Line L(26, 52) { 2, 98 }: Central Line L(2, 98) { 5, 156 }: Central Line L(5, 156) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_264 = ISOTOMIC CONJUGATE OF CIRCUMCENTER (0.5720, 0.3293) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_4 = ORTHOCENTER (0.6122, 0.3958) X_54 = KOSNITA POINT (0.5880, 0.2176) X_26 = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE (0.1382, -0.8447) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_2 = CENTROID (0.5374, 0.2000) X_98 = TARRY POINT (0.8891, -0.2281) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) A B C X264 X6 X25 X3 X49 X4 X54 X26 X52 X2 X98 X5 X156 X184 (a, b, c) = (5, 6, 7) | AB = 1

X184 (INVERSE OF X125 IN THE BROCARD CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{2}\cos A : :\)
Barycentrics: \(a^{3}\cos A : :\)
Search Value (6, 9, 13): \(-5.548868207982\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sin A \sin 2A : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines25 lines

{2, 98}, {3, 49}, {4, 54}, {5, 156}, {6, 25}, {23, 576}, {24, 389}, {26, 52}, {22, 511}, {31, 604}, {32, 211}, {48, 212}, {55, 215}, {157, 570}, {160, 571}, {199, 573}, {205, 213}, {251, 263}, {351, 686}, {381, 567}, {397, 463}, {398, 462}, {418, 577}, {572, 1011}, {647, 878}

▸ Functional Relations24 relations

Notes

X184 is the subject of Hyacinthos messages 5423-5441 (May, 2002). In #5423, Alexei Myakishev notes that X184 serves as a common vertex of three triangles inside ABC, mutually congruent and similar to ABC. (The triangles can be labeled XBCCB, XCAAC, XABBA, with BC and CB on side BC, CA and AC on side CA, and AB and BA on side AB.) See

In #5435, Paul Yiu cites Fred Lang’s construction of X184 and notes that the three triangles are then easily constructed from X184. The triangles determine three other triangles with common vertex X184; in #5437, Nikos Dergiades notes that the vertex angles of these are 4A - π, 4B - π, 4C - π, and that

if ABC is acute, then X184 = X63-of-the-orthic-triangle = X226-of-the-tangential-triangle

Randy Hutson notes that X184 is the exsimilicenter of the circumcircle and sine-triple-angle circle. (December 14, 2014)

Let A'B'C' be the intersections, other than X3, of the X3-cevians and the Brocard circle. Let A″B″C″ be the intersections, other than X6, of the X6-cevians and the Brocard circle. Then A'B'C' and A″B″C″ are perspective at X184. Also, X184 = U∩V, where U = isotomic conjugate of polar conjugate of Brocard axis (i.e., line X3X49), and V = polar conjugate of isotomic conjugate of Brocard axis (i.e., line X6X25). Let DEF be the orthic triangle. Let D' be the isotomic conjugate of X4 wrt AEF, and define E' and F' cyclically; then the lines AD', BE', CF' concur in X184. (Randy Hutson, June 1, 2015)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to ABC at X184.

In the plane of a triangle ABC, let

  • A'B'C' = tangential triangle;

  • Oa = circle with diameter BC, and define Ob and Oc cyclically;

  • Ab = A'BC'∩Oa, and define Bc and Ca cyclically;

  • Ac = B'CA'∩Oa, and define Ba and Cb cyclically;

  • A″= BcBa;∩CaCb, define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X184. (Dasari Naga Vijay Krishna, April 15, 2021)

X184 is the {X6,X25}-harmonic conjugate of X51.